Method of Estimating Modulation Transfer Function (MTF) during development of the optical system comprising Objective, Photo Detector, and Image Processor.
Any device for registration of the video information consists of the following main components (subsystems):
Since each of these subsystems contributes to the resulting resolution of the system it is possible to estimate MTF of the system using the product of MTF of the two main elements (the objective and the matrix of photo detectors) and by factoring in the influence of the digital image processing, which is usually implemented to correct the MTF of the system.
1. The estimation of the photo detectors matrix MTF.
1.1. Calculation of the frequency resolution limit for the given photo detector matrix.
Each matrix of photo detectors could be matched with the spatial frequency pass band. The limiting frequency for this pass band is Nyquist frequency defined by the Nyquist–Shannon sampling theorem. According to this theorem the exact reconstruction of a continuous-time baseband signal from its samples is possible if the signal is band limited and the sampling frequency is greater than twice the signal bandwidth. The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory").
If used matrix has the pixel size or pixel pitch , the amount of pixel per millimeter will be:
From the above equation Nyquist frequency is defined as:
Modeling pass band of the sensor matrix demand modeling the change of the sensor matrix illumination level from black level to white level and, considering pixel pitch executing the Fourier transform
is the differentiation by subtracting the adjacent dots luminance values/
is the Fourier transform.
Where is the ideal normalized MTF of the photo detectors matrix.
Also in this case only 20 dots were considered for the estimation/
1.2. The possibility of the sensor matrix MTF correction with the help of digital signal processing.
This kind of digital image processing can not make the resolution of the system greater but distinctly enhance the image quality raising the mid frequency values of the MTF. That means that digital image processing may significantly increase the level of MTF for the contrast 0.5.
2. The estimation of the required objective MTF.
The resulting MTF is greatly influenced by the two components – objective and sensor matrix.
In this article the sharp transition of luminance from the black level to the white level is investigated. Differentiating this transition leads to Line Spread Function (LSF) and the following execution of Fourier transform produces the MTF of the given system. LSF is more convenient for testing because it allows to use one dimensional calculation instead of two dimensional but for the circular aperture does not give the exact results.
Fig. 4. PSF of the ideal objective.
The main input in resulting MTF is produced by the two important elements - objective and CCD matrix.
In these formulae F is the aperture number.
MTF of the ideal objective is approximated using formula:
- MTF of the ideal objective;
The Figure 6 represents functional relation of the ideal objective diffraction limit based on Rayleigh criterion and F number.
The MTF of ideal objective for different F numbers is shown in Figure 7.
However the use of Rayleigh criterion is not appropriate since according to it the contrast is not equal zero for the resolution limit though, considering the eye perception, for the telescope it is giving the correct number. For estimation of optical system MTF Sparrow criterion should be used.
3. Estimation of the system resolution using its estimated MTF.
To build the MTF of the system it is necessary to multiply the objective’s MTF ant the sensor matrix MTF. Figures 9 and 10 represent the results of such multiplication for the CCD with the pixel pitch 1.75 um and the ideal objective with the aperture number 2.8 or 4.
1. H. Nyquist, "Certain topics in telegraph transmission theory," Trans. AIEE, vol. 47, pp. 617—644, Apr. 1928